Coronal mass ejections (CMEs) are the ejection of material from the solar corona. The ejected material is a plasma consisting primarily of electrons and protons, together with their associated magnetic field.
CMEs are now recognized as the primary solar driver of non-recurrent geomagnetic storms. See J. T. Gosling, “The Solar Flare Myth,” J. Geophys. Res., 98(A11), 18,937-18,949 (1993). Non-recurrent geomagnetic storms driven by CMEs are the most damaging of space-weather phenomena to both space- and terrestrial-based technological systems and civilian infrastructure. The direct economic consequences of space-weather phenomena have been estimated at about $200-$400 million a year. Strong storms can disrupt numerous commercial systems such as electric power grids, oil pipelines, polar aviation routes, global positioning systems, satellite- and long-line communication systems, navigation systems, satellite operations, as well as space tracking and on-orbit operations of Space Shuttle and International Space Station (ISS) activities. Solar energetic particles (SEPs) caused by CME shock fronts also represent a clear radiation hazard to manned space missions, especially for missions beyond the protection of the Earth's magnetosphere. Indeed, some have suggested that the dangers and unpredictability of solar eruptions may operationally constrain a manned mission to Mars. See C. Foullon, N. Crosby, and D. Heynderickx, “Toward Interplanetary Space Weather: Strategies for Manned Missions to Mars,” Space Weather, 3, S07004 (2005) [http://www.agu.org/journals/sw/swa/feature/article/?id=2004SW000134, accessed Jan. 15, 2010].
Understanding the processes which drive CME eruptions is therefore an important goal. Of even more concern is identifying and understanding the precursors of a CME eruption. Identifying the precursors that lead to solar eruptions is critical to developing predictive tools for space weather. If the precursors of a CME can be identified, it may be possible to forecast a CME eruption before it occurs so that appropriate steps may be taken to mitigate its effects.
CMEs are driven by magnetic forces. One main difference between driving mechanisms depends on whether the magnetic helicity and energy are first stored in the corona and later released by reconnection and instability or whether the helicity and Poynting fluxes are roughly concomitant with the eruption. The net free energy stored in the corona may be estimated by comparing the energy transported into the corona versus the “ground state” energy of a coronal potential magnetic field (B=−∇Φ) consistent with the normal component of vector magnetic field observed in the photosphere. See T. Kusano, T. Maeshiro, T. Yokoyama, and T. Sakurai, “Measurement of Magnetic Helicity Injection and Free Energy Loading into the Solar Corona,” Ap. J., 577:501-512 (2002); and M. K. Georgoulis and B. J. LaBonte, “Magnetic Energy and Helicity Budgets in the Active Region Solar Corona. I. Linear Force-Free Approximation.” Ap. J. 671:1034-1050 (2007). See also FIG. 1, which illustrates an increase in the free energy before the Halo CME which was observed just prior to 0600 UT on Nov. 4, 1997, as shown by the vertical dashed line shown in the figure. The timing and magnitude of the transport of magnetic helicity and energy through the photosphere provides an important discriminator between the mechanisms. Kusano's method was later shown to be inaccurate for estimating energy flux in the photosphere. See B. T. Welsch, W. P. Abbett, M. L. DeRosa, G. H. Fisher, M. K. Georgoulis, K. Kusano, D. W. Longope, B. Ravindra, and P. W. Schuck., “Tests and Comparisons of Velocity-Inversion Techniques,” Ap. J., 670:1434-1452 (2007) (hereinafter “Welsch 2007”), incorporated by reference herein in its entirety.
The magnetic helicity and Poynting flux may be estimated from photospheric velocities inferred from a sequence of vector magnetograms. See M. A. Berger and A. Ruzmaikin, “Rate of helicity production by solar rotation,” J. Geophys. Res., 105, 10481-10490 (2000); and P. Démoulin & M. A. Berger, “Magnetic Energy And Helicity Fluxes At The Photospheric Level,” Sol. Phys. 215, 203-215 (2003) (hereinafter “Démoulin & Berger”), the entirety of which are incorporated by reference herein. However, accurately estimating photospheric velocities from a sequence of images is extremely challenging because image motion is ambiguous. An “aperture problem” occurs when different velocities produce image dynamics that are indistinguishable. See, e.g., D. Marr and S. Ullman, “Directional Selectivity and its Use in Early Visual Processing,” Proceedings of the Royal Society of London. Series B, Biological Sciences, Vol. 211, No. 1183, pp. 151-180 (1981), incorporated by reference herein.
Optical flow methods solved these underdetermined or ill-posed problems having no unique velocity field solution by applying additional assumptions about flow structure or flow properties to enforce uniqueness. For example, P. W. Schuck, “Tracking Magnetic Footpoints with the Magnetic Induction Equation,” Ap. J., 646:1358-1391 (2006) (hereinafter “Schuck 2006”), the entirety of which is hereby incorporated by reference into the present disclosure, and Welsch 2007, supra, provide an overview of optical flow methods for recovering estimates of photospheric velocities from a sequence of magnetograms.
Previous methods have attempted to estimate the photospheric velocity using the normal component of the induction equation∂tBz+∇h·(BzVh−VzBh)=0,  (1)where the plasma velocity V and the magnetic fields B are decomposed into components based on a local right-handed Cartesian coordinate system, with Bz=Bz{circumflex over (z)} in a vertical direction along the z-axis and Bh=Bx{circumflex over (x)}+Byŷ along the horizontal image plane containing the x- and y-axes.
Démoulin & Berger cited above showed that the magnetic energy and helicity fluxes can be computed from the flux transport vectorF=UBz=BzVh−VzBh={circumflex over (z)}×(V×B)={circumflex over (z)}×(V⊥×B)  (2)where F is the flux transport vector, U is the horizontal magnetic footpoint velocity, also known as the flux transport velocity, and V⊥ is the plasma velocity perpendicular to the magnetic field.
Specifically, Démoulin & Berger showed that the relative helicity can be computed from the flux transport vector UBz:
                                                        ⅆ              Δ                        ⁢                                                  ⁢            H                                ⅆ            t                          =                  -                      ∫                          2              ⁢                                                A                  p                                ·                                  (                                      UB                    z                                    )                                            ⁢                              ⅆ                x                            ⁢                              ⅆ                y                                                                        (        3        )            where Ap={circumflex over (z)}×∇Φp is the potential reference field (with zero helicity) which satisfies the relation {circumflex over (z)}·(∇×Ap)=∇h2Φp=Bz and the integral is taken over all points x, y in the image plane.
Similarly, the net power through the photosphere can be computed from the horizontal magnetic field Bh and the flux transport vector UBz:
                                                        ⅆ              Δ                        ⁢                                                  ⁢            E                                ⅆ            t                          =                  -                      ∫                                                                                B                    h                                    ·                                      (                                          UB                      z                                        )                                                                    4                  ⁢                                                                          ⁢                  π                                            ⁢                              ⅆ                x                            ⁢                              ⅆ                y                                                                        (        4        )            
The net free energy ΔEf available for production of a CME may be found from the net power through the photosphere
            ⅆ      Δ        ⁢                  ⁢    E        ⅆ    t  and the time rate of change of energy in the associated potential magnetic field
      ⅆ          E      p            ⅆ    t  
                              Δ          ⁢                                          ⁢                      E            f                          =                  ∫                      ⅆ                          t              ⁡                              (                                                                                                    ⅆ                        Δ                                            ⁢                                                                                          ⁢                      E                                                              ⅆ                      t                                                        -                                                            ⅆ                                              E                        p                                                                                    ⅆ                      t                                                                      )                                                                        (        5        )            
where
            ⅆ      Δ        ⁢                  ⁢    E        ⅆ    t  is as defined above and
      ⅆ          E      p            ⅆ    t  is the ground state potential energy due to the normal component Bz of the magnetic field which may be time dependent. See, e.g., M. K. Georgoulis, supra. As noted above, the ground truth flux transport vector 1113, comprises two terms, BzVh and VzBh. These terms represent shearing due to horizontal motion of the plasma and flux emergence due to vertical motion of the magnetic field, respectively. Thus, it is desirable to know the plasma velocity V, since it can be used to obtain the horizontal footpoint velocity U, which in turn can be used to compute the free energy and helicity available for the creation of a coronal mass ejection.
It will be noted at this point that as used herein, uppercase “U” denotes the “ground truth” flux transport velocity (also known as “horizontal footpoint velocity”) and “V” denotes the ground truth plasma velocity, e.g., as modeled by anelastic magnetohydrodynamic (ANMHD) simulations or otherwise, see W. P. Abbett, G. H. Fisher, and Y. Fan, “The Three-Dimensional Evolution Of Rising, Twisted Magnetic Flux Tubes In A Gravitationally Stratified Model Convection Zone,” Ap. J., 540:548-562 (2000); W. P. Abbett, G. H. Fisher, Y. Fan, and D. J. Berick, “The Dynamic Evolution Of Twisted Magnetic Flux Tubes In A Three-Dimensional Convecting Flow. II. Turbulent Pumping And The Cohesion Of S2-Loops,” Ap. J., 612:557-575 (2004), incorporated by reference herein in their entirety, while lowercase “u” and “v” denote the corresponding estimated velocities in accordance with the present invention; see P. W. Schuck (2006), supra, and B. T. Welsch et al. (2007), supra,
Démoulin & Berger argued, based on the geometry of a magnetic field line passing through the photosphere, that the term UBz could be substituted for (BzVh−VzBh) in Equation (1), resulting in the following continuity equation for the vertical magnetic field∂tBz+∇h·(UBz)=0  (6)which could be solved directly for UBz without the need for knowledge of the plasma velocity components Vh and Vz.
Démoulin & Berger further argued that existing tracking methods such as local correlation tracking described in L. November and G. Simon, “Precise Proper-Motion Measurement of Solar Granulation,” Ap. J., 333:427-442 (1988) provided an estimate u of the ground truth magnetic footpoint velocity U and that therefore Equation (3) can then be rewritten as∂tBz+∇h·(uBz)=0  (7)
Based on Démoulin & Berger's arguments described above, attempts were made to estimate the horizontal magnetic footpoint velocity u by solving Equation (7) for u using a differential affine velocity estimator (DAVE) given a sequence of Bz images. See Schuck 2006 and Welsch 2007, supra.
However, Equation (7) above contains only a Bz term and does not include any information about the horizontal components Bx and By of the magnetic field. When the values of u obtained from Equation (7) were tested against ground truth values of U such as those obtained from ANMHD simulations, it was found that such estimated values of diverged significantly from the ground truth values. See Welsch 2007 and Schuck 2006, supra; see also P. W. Schuck, “Tracking Vector Magnetograms With The Magnetic Induction Equation,” Ap. J. 683:1134-1152 (2008) (hereinafter “Schuck 2008”), incorporated herein by reference in its entirety. See also FIGS. 2A and 2B which show the x and y components of the ground truth velocity flux transport vector UBz versus the flux transport vector uBz based on the estimated footpoint velocity u computed using Equation (7). As can be seen in FIGS. 2A and 2B, the estimated flux transport vectors uxBz and uyBz diverge significantly from the ground truth values determined directly from the simulation data. Thus, the prior methods for estimating u and uBz did not provide the desirable level of accuracy needed for prediction of CMEs, and there remained a need for a method of accurately estimating the plasma velocities vh and vz so that more accurate values of u and uBz could be obtained.